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Simulating diffusion processes in discontinuous media: benchmark tests. (English) Zbl 1349.65019
Summary: We present several benchmark tests for Monte Carlo methods simulating diffusion in one-dimensional discontinuous media. These benchmark tests aim at studying the potential bias of the schemes and their impact on the estimation of micro- or macroscopic quantities (repartition of masses, fluxes, mean residence time,\(\ldots\)). These benchmark tests are backed by a statistical analysis to filter out the bias from the unavoidable Monte Carlo error. We apply them on four different algorithms. The results of the numerical tests give a valuable insight into the fine behavior of these schemes, as well as rules to choose between them.

MSC:
65C05 Monte Carlo methods
60J60 Diffusion processes
Software:
COUPLEX
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[1] Ackerer, P.; Mose, R., Comment on “diffusion theory for transport in porous media: transition-probability densities of diffusion processes corresponding to advection-dispersion equations” by eric M. labolle et al., Water Resour. Res., 36, 3, 819-821, (2000)
[2] Appuhamillage, T. A.; Bokil, V. A.; Thomann, E.; Waymire, E.; Wood, B. D., Solute transport across an interface: a Fickian theory for skewness in breakthrough curves, Water Resour. Res., 46, (2010)
[3] Appuhamillage, T. A.; Bokil, V. A.; Thomann, E.; Waymire, E.; Wood, B. D., Occupation and local times for skew Brownian motion with application to dispersion across an interface, Ann. Appl. Probab., 21, 1, 183-214, (2011) · Zbl 1226.60113
[4] Bechtold, M.; Vanderborght, J.; Ippisch, O.; Vereecken, H. V., Efficient random walk particle tracking algorithm for advective-dispersive transport in media with discontinuous dispersion coefficients and water contents, Water Resour. Res., 47, (2011)
[5] Berezhkovskii, A. M.; Zaloj, V.; Agmon, N., Residence time distribution of a Brownian particle, Phys. Rev. E, 57, 4, 3937-3947, (1998)
[6] Bourgeat, A.; Kern, M.; Schumacher, S.; Talandier, J., The \sccouplex test cases: nuclear waste disposal simulation, Comput. Geosci., 8, 83-98, (2004) · Zbl 1060.86002
[7] Bossy, M.; Champagnat, N.; Maire, S.; Talay, D., Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics, ESAIM M2AN, 44, 5, 997-1048, (2010) · Zbl 1204.82020
[8] Cantrell, R. S.; Cosner, C., Diffusion models for population dynamics incorporating individual behavior at boundaries: applications to refuge design, Theor. Popul. Biol., 55, 2, 189-207, (1999) · Zbl 0958.92028
[9] Cortis, A.; Zoia, A., Model of dispersive transport across sharp interfaces between porous materials, Phys. Rev. E, 80, (2009)
[10] Cox, D. R.; Hinkley, D. V., Theoretical statistics, (1974), Chapman and Hall London · Zbl 0334.62003
[11] Delay, F.; Ackerer, Ph.; Danquigny, C., Simulating solute transport in porous or fractured formations using random walks particle tracking: a review, Vadose Zone J., 4, 360-379, (2005)
[12] Étoré, P., On random walk simulation of one-dimensional diffusion processes with discontinuous coefficients, Electron. J. Probab., 11, 9, 249-275, (2006) · Zbl 1112.60061
[13] Étoré, P.; Lejay, A., A donsker theorem to simulate one-dimensional processes with measurable coefficients, ESAIM Probab. Stat., 11, 301-326, (2007) · Zbl 1181.60123
[14] Étoré, P.; Martinez, M., Exact simulation of one-dimensional stochastic differential equations involving the local time at zero of the unknown process, Monte Carlo Methods Appl., 19, 1, (2013) · Zbl 1269.65007
[15] Feller, W., On the Kolmogorov-Smirnov limit theorems for empirical distributions, Ann. Math. Stat., 19, 177-189, (1948) · Zbl 0032.03801
[16] Fieremans, E.; Novikov, D. S.; Jensen, J. H.; Helpern, J. A., Monte Carlo study of a two-compartment exchange model of diffusion, NMR Biomed., 23, 711-724, (2010)
[17] Fisz, M., Probability theory and mathematical statistics, (1963), John Wiley & Sons Inc. New York · Zbl 0123.34504
[18] Gardiner, C., Stochastic methods. A handbook for the natural and social sciences, Springer Series in Synergetics, (2009), Springer-Verlag Berlin · Zbl 1181.60001
[19] Gräwe, U.; Deleersnijder, E.; Shah, S. H.A. M.; Heemink, A. W., Why the Euler scheme in particle tracking is not enough: the shallow-sea pycnocline test case, Ocean Dyn., 62, 4, 501-514, (2012)
[20] Gräwe, U., Implementation of high-order particle-tracking schemes in a water column model, Ocean Model., 36, 1-2, 80-89, (2011)
[21] de Haan, H. W.; Chubynsky, M.; Slater, G. W., Monte Carlo approaches for simulating a particle at a diffusivity interface and the “ito-Stratonovich dilemma”, (2012-08-24)
[22] Hoteit, H.; Mose, R.; Younes, A.; Lehmann, F.; Ackerer, Ph., Three-dimensional modeling of mass transfer in porous media using the mixed hybrid finite elements and the random-walk methods, Math. Geol., 34, 4, 435-456, (2002) · Zbl 1107.76401
[23] Kloeden, P. E.; Platen, E., Numerical solution of stochastic differential equations, Applications of Mathematics (New York), vol. 23, (1992), Springer-Verlag Berlin · Zbl 0925.65261
[24] LaBolle, E. M.; Zhang, Y., Reply to comment by D.-H. lim on “diffusion processes in composite porous media and their numerical integration by random walks: generalized stochastic differential equations with discontinuous coefficients”, Water Resour. Res., 42, (2006)
[25] LaBolle, E. M.; Quastel, J.; Fogg, G. E.; Gravner, J., Diffusion processes in composite porous media and their numerical integration by random walks: generalized stochastic differential equations with discontinuous coefficients, Water Resour. Res., 36, 651-662, (2000)
[26] LaBolle, E. M.; Fogg, G. E.; Thomson, A. F.B., Diffusion theory for transport in porous media: transition-probability densities of diffusion processes corresponding to advection-dispersion equations, Water Resour. Res., 34, 7, 1685-1693, (1998)
[27] LaBolle, E. M.; Fogg, G. E.; Thomson, A. F.B., Random-walk simulation of transport in heterogeneous porous media: local mass-conservation problem and implementation methods, Water Resour. Res., 32, 3, 582-593, (1996)
[28] Lejay, A.; Pichot, G., Simulating diffusion processes in discontinuous media: a numerical scheme with constant time steps, J. Comput. Phys., 231, 21, 7299-7314, (2012) · Zbl 1284.65007
[29] Lejay, A., On the constructions of the skew Brownian motion, Probab. Surv., 3, 413-466, (2006) · Zbl 1189.60145
[30] Lejay, A.; Martinez, M., A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients, Ann. Appl. Probab., 16, 1, 107-139, (2006) · Zbl 1094.60056
[31] Lejay, A., Simulating a diffusion on a graph. application to reservoir engineering, Monte Carlo Methods Appl., 9, 3, 241-256, (2003) · Zbl 1072.76062
[32] Lejay, A.; Maire, S., Computing the principal eigenvalue of the Laplace operator by a stochastic method, Math. Comput. Simul., 73, 3, 351-363, (2007) · Zbl 1110.65105
[33] Lim, D.-H., Comment on “diffusion processes in composite porous media and their numerical integration by random walks: generalized stochastic differential equations with discontinuous coefficients” by E.M. labolle, J. quastel, G.E. fogg, J. gravner, Water Resour. Res., 42, (2006)
[34] Marcowith, A.; Casse, F., Postshock turbulence and diffusive shock acceleration in Young supernova remnants, Astron. Astrophys., 515, A90, (2010)
[35] Marseguerra, M.; Zoia, A., Normal and anomalous transport across an interface: Monte Carlo and analytical approach, Ann. Nucl. Energy, 33, 17-18, 1396-1407, (2006)
[36] Martinez, M., Interprétations probabilistes d’opérateurs sous forme divergence et analyse de méthodes numériques associées, (2004), Université de Provence/INRIA Sophia-Antipolis, PhD thesis
[37] Martinez, M.; Talay, D., Discrétisation d’équations différentielles stochastiques unidimensionnelles à générateur sous forme divergence avec coefficient discontinu, C. R. Math. Acad. Sci. Paris, 342, 1, 51-56, (2006) · Zbl 1082.60514
[38] Martinez, M.; Talay, D., One-dimensional parabolic diffraction equations: pointwise estimates and discretization of related stochastic differential equations with weighted local times, Electron. J. Probab., 17, 27, (2012) · Zbl 1244.60058
[39] Mascagni, M.; Simonov, N. A., Monte Carlo methods for calculating some physical properties of large molecules, SIAM J. Sci. Comput., 26, 1, 339-357, (2004) · Zbl 1075.65003
[40] Milstein, G. N.; Tretyakov, M. V., Stochastic numerics for mathematical physics, Scientific Computation, (2004), Springer-Verlag Berlin · Zbl 1085.60004
[41] Øksendal, B., Stochastic differential equations: an introduction with applications, Universitext, (2003), Springer-Verlag Berlin · Zbl 1025.60026
[42] Ovaskainen, O.; Cornell, S. J., Biased movement at a boundary and conditional occupancy times for diffusion processes, J. Appl. Probab., 40, 3, 557-580, (2003) · Zbl 1078.60061
[43] Ramirez, J. M.; Thomann, E. A.; Waymire, E. C., Advection-dispersion across interfaces, Stat. Sci., 28, 4, 487-509, (2013) · Zbl 1331.60003
[44] Ramirez, J.; Thomann, E.; Waymire, E.; Chastenet, J.; Wood, B., A note on the theoretical foundations of particle tracking methods in heterogeneous porous media, Water Resour. Res., 44, (2007)
[45] Ramirez, J. M., Skew Brownian motion and branching processes applied to diffusion-advection in heterogenous media and fluid flow, (2007), Oregon State University, PhD thesis
[46] Ramirez, J. M.; Thomann, E. A.; Waymire, E. C.; Haggerty, R.; Wood, B., A generalized Taylor-aris formula and skew diffusion, Multiscale Model. Simul., 5, 3, 786-801, (2006) · Zbl 1122.60072
[47] Salamon, P.; Fernàndez-Garcia, D.; Gómez-Hernández, J. J., A review and numerical assessment of the random walk particle tracking method, J. Contam. Hydrol., 87, 3-4, 277-305, (2006)
[48] Semra, K., Modélisation tridimensionnelle du transport d’un traceur en milieux poreux saturé: évaluation des théories stochastiques, (1994), Université Louis Pasteur Strasbourg, PhD thesis
[49] Sherman, R., Error of the normal approximation to the sum of N random variables, Biometrika, 58, (1971) · Zbl 0226.62037
[50] Shorack, G.; Wellner, J., Empirical processes with applications to statistics, (1986), John Wiley & Sons · Zbl 1170.62365
[51] Smirnov, N., Table for estimating the goodness of fit of empirical distributions, Ann. Math. Stat., 19, 279-281, (1948) · Zbl 0031.37001
[52] Spivakovskaya, D.; Heemink, A. W.; Deleersnijder, E., Lagrangian modelling of multi-dimensional advection-diffusion with space-varying diffusivities: theory and idealized test cases, Ocean Dyn., 57, 3, 189-203, (2007)
[53] Spivakovskaya, D.; Heemink, A. W.; Deleersnijder, E., The backward ito method for the Lagrangian simulation of transport processes with large space variations of the diffusivity, Ocean Sci., 3, 4, 525-535, (2007)
[54] Thomson, D. J.; Physick, W. L.; Maryon, R. H., Treatment of interfaces in random walk dispersion models, J. Appl. Meteorol., 36, 1284-1295, (1997)
[55] Uffink, G. J.M., Analysis of dispersion by the random walk method, (1990-02-06), Delft University The Netherlands, PhD thesis
[56] Zhang, M., Calculation of diffusive shock acceleration of charged particles by skew Brownian motion, Astrophys. J., 541, 428-435, (2000)
[57] Zheng, C.; Bennett, G., Applied contaminant transport modeling, (2002), Wiley-Interscience
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